A) \[~{{x}^{2}}-x-1=0\]
B) \[~{{x}^{2}}-\text{ }x+1=0\]
C) \[~{{x}^{2}}+x-1=0\]
D) \[~{{x}^{2}}+\text{ }x+1=0\]
Correct Answer: D
Solution :
Given equation is \[{{x}^{2}}+x+1=0\] \[\Rightarrow \] \[x=\frac{-1\pm \sqrt{1-4}}{2}\] \[=\frac{-1\pm \sqrt{3}i}{2}\] \[\Rightarrow \]\[\alpha =\frac{-1+\sqrt{3}\,i}{2}=\omega ,\beta =\frac{-1-\sqrt{3}i}{2}={{\omega }^{2}}\] Now, \[{{\alpha }^{19}}={{\omega }^{19}}=\omega \] and \[{{\beta }^{7}}={{({{\omega }^{2}})}^{7}}={{\omega }^{2}}\] Since, this will be same as the roots of the given equation. \[\therefore \]Required equation is \[{{x}^{2}}+x+1=0\]
You need to login to perform this action.
You will be redirected in
3 sec
